How Quantum Physics Broke the Laws of Statistics
Statistics is a core pillar of Data Science, yet its assumptions are not always fully tested. This is exacerbated by the rise of quantum computing, where even statistical axioms can be violated. In this article, we explore just how quantum physics breaks statistics, and uncover ways to understand it using data science analogies.
Let's play a coin-toss game: toss three coins, and try to have them all land differently. This is a seemingly impossible task, because no matter how rigged a coin is, it can only have two sides. There simply aren't enough possibilities for all three tosses to land differently.
Yet, with the power of quantum Physics, such an impossible feat can be achieved statistically: three coin tosses can all land differently. And the reward for winning? 2022's Nobel Prize in Physics, which was awarded to Alain Aspect, John Clauser, and Anton Zeilinger on 2022-10-04.
According to nobelprize.org, their achievements were
"for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science."
This sentence is filled with jargon: entangled photons, Bell inequalities, and quantum information science. We need a simpler, plain English description for such an important feat. Here's a translation:
Scientists showed that our statistical view of the world is flawed, by showing that quantum physics can defy seemingly impossible odds.
The details of these impossible odds are captured by mathematical formulae called Bell inequalities. Instead of flipping coins, researchers demonstrated these impossible odds by playing with lasers (using beams of entangled photons).
How is this relevant to data science? Since our quantum mechanical world is the ultimate source of data, flaws in our statistical laws could disrupt the very foundation of data science. If Statistics is indeed incomplete, we wouldn't be able to trust conclusions derived from it.
Fortunately, in our Universe, these statistical flaws tend to be very tiny and negligible. Nevertheless, it is important to understand how classical statistics needs to be modified, as data science in the distant future may need to incorporate these flaws (e.g., in quantum computers).
Before answering how quantum physics defies the laws of statistics, we first need to understand how statistics works as an effective description for our world.
Trading Unpredictability for Probability
Flip a coin, you get heads/tails. Yet coins aren't exactly random: A robot with perfect control can seriously rig a coin-toss.
What does a 50/50 probability mean? A coin's orientation is very sensitive to the minute details of its surrounding. This makes it difficult to predict a coin's landing orientation. So instead of solving very complicated equations to come up with a deterministic outcome, we opt for a nondeterministic one. How? We observe that typical coins are pretty symmetrical with respect to heads/tails. In the absence of any particular bias, 50/50 odds would be a great approximation (although studies have shown these odds can be altered, e.g., Clark MP et al.).
To summarize,
Probabilities are approximations for modeling details of a complex system. Complicated physics is traded for uncertainties in order to simplify the mathematics.
From weather patterns to economics and healthcare, uncertainties can be traced back to complex dynamics. Mathematicians have converted these approximations into rigorous theorems based on axioms, to help us manipulate and derive insights from unpredictable outcomes.
Counting Probabilities
How does quantum physics break the laws of statistics? It violates the Additivity Axiom.
How does this Axiom work? Let's consider some common scenarios where we use statistics to make decisions:
- When it's rainy
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